6 research outputs found
An Internal Language for Categories Enriched over Generalised Metric Spaces
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale ?, which covers the cases of (in)equations and (ultra)metric equations among others.
Our main result is the introduction of a ?-equational deductive system for linear ?-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces.
We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour
Categorical Models for a Semantically Linear Lambda-calculus
This paper is about a categorical approach to model a very simple
Semantically Linear lambda calculus, named Sll-calculus. This is a core
calculus underlying the programming language SlPCF. In particular, in this
work, we introduce the notion of Sll-Category, which is able to describe a very
large class of sound models of Sll-calculus. Sll-Category extends in the
natural way Benton, Bierman, Hyland and de Paiva's Linear Category, in order to
soundly interpret all the constructs of Sll-calculus. This category is general
enough to catch interesting models in Scott Domains and Coherence Spaces
Partial Recursive Functions and Finality
Abstract. We seek universal categorical conditions ensuring the representability of all partial recursive functions. In the category Pfn of sets and partial functions, the natural numbers provide both an initial algebra and a final coalgebra for the functor 1 + −. We recount how finality yields closure of the partial functions on natural numbers under Kleene’s µ-recursion scheme. Noting that Pfn is not cartesian, we then build on work of Paré and Román, obtaining weak initiality and finality conditions on natural numbers algebras in monoidal categories that ensure the (weak) representability of all partial recursive functions. We further obtain some positive results on strong representability. All these results adapt to Kleisli categories of cartesian categories with natural numbers algebras. However, in general, not all partial recursive functions need be strongly representable.